<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC
 "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN"
 "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>pi-qmc: Harmonic Oscillator Example</title>
<meta charset="UTF-8" />
<link rel="stylesheet" href="pi.css" type="text/css"/>
<script type="text/javascript" src="pagecontents.js"></script>
</head>

<body>
<div class="nav">
<a href="Overview.xhtml" class="prev">Overview</a>
<a href="Introduction.xhtml" class="up">Introduction</a>
</div>
<h1>Example: A Simple Harmonic Oscillator</h1>
<h2>Background</h2>
<h3>Eigenstates of the Simple Harmonic Oscillator</h3>
<h3>The Classical Partition Function</h3>
<h3>The Quantum Mechanical Partition Function</h3>
<p>In one dimension, the partition function of the simple harmonic
oscillator is
<div class="math">Z = Σ<sub>n=0</sub><sup>∞</sup> exp[-βℏω(n+½)]
= [2 cosh(βℏω/2)]<sup>-1</sup>.</div>
For <span div="math">N</span> oscillators in 
<span div="math">D</span> dimensions, the partition function is
<div class="math">Z = [2 cosh(βℏω/2)]<sup>-ND</sup>.</div>
The Helmholtz free energy is
<div class="math">F = -k<sub>B</sub>T ln(Z)
= ND k<sub>B</sub>T ln[2 cosh(βℏω/2)].</div>
The total energy is
<div class="math">E = -(d/dβ) ln(Z)
= ND (½ℏω) coth(βℏω/2).</div>
</p>
<h3>The Density Matrix</h3>
<h2>Simulating a Simple Harmonic Oscillator</h2>
<h3>Setting up the Input file</h3>
<h3>Calculating the Energy</h3>
<h3>Calculating the Density</h3>
<h3>Calculating the Polarizability</h3>

</body>
</html>

